# Properties

 Label 22.2.c.a Level $22$ Weight $2$ Character orbit 22.c Analytic conductor $0.176$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [22,2,Mod(3,22)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(22, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([8]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("22.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$22 = 2 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 22.c (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.175670884447$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{10} q^{2} + ( - \zeta_{10}^{3} + \zeta_{10} - 1) q^{3} + \zeta_{10}^{2} q^{4} + (2 \zeta_{10}^{3} - 2) q^{5} + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + \cdots - 1) q^{6} + \cdots + (3 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{9} +O(q^{10})$$ q - z * q^2 + (-z^3 + z - 1) * q^3 + z^2 * q^4 + (2*z^3 - 2) * q^5 + (z^3 - 2*z^2 + 2*z - 1) * q^6 - 2*z^2 * q^7 - z^3 * q^8 + (3*z^2 - 2*z + 3) * q^9 $$q - \zeta_{10} q^{2} + ( - \zeta_{10}^{3} + \zeta_{10} - 1) q^{3} + \zeta_{10}^{2} q^{4} + (2 \zeta_{10}^{3} - 2) q^{5} + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + \cdots - 1) q^{6} + \cdots + ( - 13 \zeta_{10}^{3} + 8 \zeta_{10}^{2} + \cdots + 3) q^{99} +O(q^{100})$$ q - z * q^2 + (-z^3 + z - 1) * q^3 + z^2 * q^4 + (2*z^3 - 2) * q^5 + (z^3 - 2*z^2 + 2*z - 1) * q^6 - 2*z^2 * q^7 - z^3 * q^8 + (3*z^2 - 2*z + 3) * q^9 + (-2*z^3 + 2*z^2 + 2) * q^10 + (-3*z^3 + 2*z^2 - 4*z + 2) * q^11 + (z^3 - z^2 + 1) * q^12 + (-2*z^2 + 2*z - 2) * q^13 + 2*z^3 * q^14 + (2*z^3 - 2*z^2 + 2*z) * q^15 + (z^3 - z^2 + z - 1) * q^16 + (-z^2 + z) * q^17 + (-3*z^3 + 2*z^2 - 3*z) * q^18 + (4*z^3 + 3*z - 3) * q^19 + (-2*z^2 - 2) * q^20 + (-2*z^3 + 2*z^2 - 2) * q^21 + (z^3 + z^2 + z - 3) * q^22 + (-2*z^3 + 2*z^2) * q^23 + (z^2 - 2*z + 1) * q^24 + (-3*z^3 - 4*z + 4) * q^25 + (2*z^3 - 2*z^2 + 2*z) * q^26 + (-z^3 - z^2 + z + 1) * q^27 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^28 + (4*z^3 - 2*z^2 + 4*z) * q^29 + (-2*z + 2) * q^30 - 2*z * q^31 + q^32 + (4*z^3 - 7*z^2 + 4*z - 1) * q^33 + (z^3 - z^2) * q^34 + (4*z^2 + 4) * q^35 + (z^3 + 3*z - 3) * q^36 + (-6*z^3 + 6*z^2 - 6*z) * q^37 + (-4*z^3 + z^2 - z + 4) * q^38 + (-2*z^3 + 6*z^2 - 6*z + 2) * q^39 + (2*z^3 + 2*z) * q^40 + (-5*z^3 - z + 1) * q^41 + (-2*z^2 + 4*z - 2) * q^42 + (9*z^3 - 9*z^2 - 3) * q^43 + (-2*z^3 + 2*z + 1) * q^44 + (2*z^3 - 2*z^2 - 8) * q^45 + (-2*z^2 + 2*z - 2) * q^46 + (4*z^3 + 4*z - 4) * q^47 + (-z^3 + 2*z^2 - z) * q^48 + (-3*z^3 + 3*z^2 - 3*z + 3) * q^49 + (3*z^3 + z^2 - z - 3) * q^50 + (-2*z^3 + 3*z^2 - 2*z) * q^51 + (-2*z + 2) * q^52 + (-4*z^2 + 8*z - 4) * q^53 + (2*z^3 - 2*z^2 - 1) * q^54 + (2*z^3 + 4*z^2 + 6*z) * q^55 - 2 * q^56 + (2*z^2 - z + 2) * q^57 + (-2*z^3 - 4*z + 4) * q^58 + (-z^3 - 7*z^2 - z) * q^59 + (2*z^2 - 2*z) * q^60 + (-4*z^2 + 4*z) * q^61 + 2*z^2 * q^62 + (-2*z^3 - 6*z + 6) * q^63 - z * q^64 + 4 * q^65 + (3*z^3 - 3*z + 4) * q^66 + (-5*z^3 + 5*z^2 + 8) * q^67 + (z^2 - z + 1) * q^68 + (2*z^3 - 4*z + 4) * q^69 + (-4*z^3 - 4*z) * q^70 + (-4*z^3 + 2*z^2 - 2*z + 4) * q^71 + (-z^3 - 2*z^2 + 2*z + 1) * q^72 + (-z^3 + 12*z^2 - z) * q^73 + (6*z - 6) * q^74 + (-5*z^2 + 6*z - 5) * q^75 + (3*z^3 - 3*z^2 - 4) * q^76 + (4*z^3 - 4*z - 2) * q^77 + (-4*z^3 + 4*z^2 - 2) * q^78 + (-12*z^2 + 6*z - 12) * q^79 + (-2*z^3 - 2*z + 2) * q^80 + (6*z^3 - 2*z^2 + 6*z) * q^81 + (5*z^3 - 4*z^2 + 4*z - 5) * q^82 + (5*z^3 + 2*z^2 - 2*z - 5) * q^83 + (2*z^3 - 4*z^2 + 2*z) * q^84 + 2*z^3 * q^85 + (9*z^2 - 6*z + 9) * q^86 + (-6*z^3 + 6*z^2 - 2) * q^87 + (2*z^3 - 4*z^2 + z - 2) * q^88 + (5*z^3 - 5*z^2 - 5) * q^89 + (2*z^2 + 6*z + 2) * q^90 + (4*z - 4) * q^91 + (2*z^3 - 2*z^2 + 2*z) * q^92 + (2*z^3 - 4*z^2 + 4*z - 2) * q^93 + (-4*z^3 + 4) * q^94 + (-8*z^3 - 6*z^2 - 8*z) * q^95 + (-z^3 + z - 1) * q^96 + (3*z^2 - 12*z + 3) * q^97 - 3 * q^98 + (-13*z^3 + 8*z^2 - 4*z + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} - 4 q^{3} - q^{4} - 6 q^{5} + q^{6} + 2 q^{7} - q^{8} + 7 q^{9}+O(q^{10})$$ 4 * q - q^2 - 4 * q^3 - q^4 - 6 * q^5 + q^6 + 2 * q^7 - q^8 + 7 * q^9 $$4 q - q^{2} - 4 q^{3} - q^{4} - 6 q^{5} + q^{6} + 2 q^{7} - q^{8} + 7 q^{9} + 4 q^{10} - q^{11} + 6 q^{12} - 4 q^{13} + 2 q^{14} + 6 q^{15} - q^{16} + 2 q^{17} - 8 q^{18} - 5 q^{19} - 6 q^{20} - 12 q^{21} - 11 q^{22} - 4 q^{23} + q^{24} + 9 q^{25} + 6 q^{26} + 5 q^{27} + 2 q^{28} + 10 q^{29} + 6 q^{30} - 2 q^{31} + 4 q^{32} + 11 q^{33} + 2 q^{34} + 12 q^{35} - 8 q^{36} - 18 q^{37} + 10 q^{38} - 6 q^{39} + 4 q^{40} - 2 q^{41} - 2 q^{42} + 6 q^{43} + 4 q^{44} - 28 q^{45} - 4 q^{46} - 8 q^{47} - 4 q^{48} + 3 q^{49} - 11 q^{50} - 7 q^{51} + 6 q^{52} - 4 q^{53} + 4 q^{55} - 8 q^{56} + 5 q^{57} + 10 q^{58} + 5 q^{59} - 4 q^{60} + 8 q^{61} - 2 q^{62} + 16 q^{63} - q^{64} + 16 q^{65} + 16 q^{66} + 22 q^{67} + 2 q^{68} + 14 q^{69} - 8 q^{70} + 8 q^{71} + 7 q^{72} - 14 q^{73} - 18 q^{74} - 9 q^{75} - 10 q^{76} - 8 q^{77} - 16 q^{78} - 30 q^{79} + 4 q^{80} + 14 q^{81} - 7 q^{82} - 19 q^{83} + 8 q^{84} + 2 q^{85} + 21 q^{86} - 20 q^{87} - q^{88} - 10 q^{89} + 12 q^{90} - 12 q^{91} + 6 q^{92} + 2 q^{93} + 12 q^{94} - 10 q^{95} - 4 q^{96} - 3 q^{97} - 12 q^{98} - 13 q^{99}+O(q^{100})$$ 4 * q - q^2 - 4 * q^3 - q^4 - 6 * q^5 + q^6 + 2 * q^7 - q^8 + 7 * q^9 + 4 * q^10 - q^11 + 6 * q^12 - 4 * q^13 + 2 * q^14 + 6 * q^15 - q^16 + 2 * q^17 - 8 * q^18 - 5 * q^19 - 6 * q^20 - 12 * q^21 - 11 * q^22 - 4 * q^23 + q^24 + 9 * q^25 + 6 * q^26 + 5 * q^27 + 2 * q^28 + 10 * q^29 + 6 * q^30 - 2 * q^31 + 4 * q^32 + 11 * q^33 + 2 * q^34 + 12 * q^35 - 8 * q^36 - 18 * q^37 + 10 * q^38 - 6 * q^39 + 4 * q^40 - 2 * q^41 - 2 * q^42 + 6 * q^43 + 4 * q^44 - 28 * q^45 - 4 * q^46 - 8 * q^47 - 4 * q^48 + 3 * q^49 - 11 * q^50 - 7 * q^51 + 6 * q^52 - 4 * q^53 + 4 * q^55 - 8 * q^56 + 5 * q^57 + 10 * q^58 + 5 * q^59 - 4 * q^60 + 8 * q^61 - 2 * q^62 + 16 * q^63 - q^64 + 16 * q^65 + 16 * q^66 + 22 * q^67 + 2 * q^68 + 14 * q^69 - 8 * q^70 + 8 * q^71 + 7 * q^72 - 14 * q^73 - 18 * q^74 - 9 * q^75 - 10 * q^76 - 8 * q^77 - 16 * q^78 - 30 * q^79 + 4 * q^80 + 14 * q^81 - 7 * q^82 - 19 * q^83 + 8 * q^84 + 2 * q^85 + 21 * q^86 - 20 * q^87 - q^88 - 10 * q^89 + 12 * q^90 - 12 * q^91 + 6 * q^92 + 2 * q^93 + 12 * q^94 - 10 * q^95 - 4 * q^96 - 3 * q^97 - 12 * q^98 - 13 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/22\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$\chi(n)$$ $$\zeta_{10}^{2}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.809017 − 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 + 0.587785i
−0.809017 + 0.587785i 0.118034 + 0.363271i 0.309017 0.951057i −2.61803 1.90211i −0.309017 0.224514i −0.618034 + 1.90211i 0.309017 + 0.951057i 2.30902 1.67760i 3.23607
5.1 0.309017 + 0.951057i −2.11803 1.53884i −0.809017 + 0.587785i −0.381966 + 1.17557i 0.809017 2.48990i 1.61803 1.17557i −0.809017 0.587785i 1.19098 + 3.66547i −1.23607
9.1 0.309017 0.951057i −2.11803 + 1.53884i −0.809017 0.587785i −0.381966 1.17557i 0.809017 + 2.48990i 1.61803 + 1.17557i −0.809017 + 0.587785i 1.19098 3.66547i −1.23607
15.1 −0.809017 0.587785i 0.118034 0.363271i 0.309017 + 0.951057i −2.61803 + 1.90211i −0.309017 + 0.224514i −0.618034 1.90211i 0.309017 0.951057i 2.30902 + 1.67760i 3.23607
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.2.c.a 4
3.b odd 2 1 198.2.f.e 4
4.b odd 2 1 176.2.m.c 4
5.b even 2 1 550.2.h.h 4
5.c odd 4 2 550.2.ba.c 8
8.b even 2 1 704.2.m.h 4
8.d odd 2 1 704.2.m.a 4
11.b odd 2 1 242.2.c.c 4
11.c even 5 1 inner 22.2.c.a 4
11.c even 5 1 242.2.a.f 2
11.c even 5 2 242.2.c.a 4
11.d odd 10 1 242.2.a.d 2
11.d odd 10 1 242.2.c.c 4
11.d odd 10 2 242.2.c.d 4
33.f even 10 1 2178.2.a.x 2
33.h odd 10 1 198.2.f.e 4
33.h odd 10 1 2178.2.a.p 2
44.g even 10 1 1936.2.a.n 2
44.h odd 10 1 176.2.m.c 4
44.h odd 10 1 1936.2.a.o 2
55.h odd 10 1 6050.2.a.ci 2
55.j even 10 1 550.2.h.h 4
55.j even 10 1 6050.2.a.bs 2
55.k odd 20 2 550.2.ba.c 8
88.k even 10 1 7744.2.a.cy 2
88.l odd 10 1 704.2.m.a 4
88.l odd 10 1 7744.2.a.cz 2
88.o even 10 1 704.2.m.h 4
88.o even 10 1 7744.2.a.bm 2
88.p odd 10 1 7744.2.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.2.c.a 4 1.a even 1 1 trivial
22.2.c.a 4 11.c even 5 1 inner
176.2.m.c 4 4.b odd 2 1
176.2.m.c 4 44.h odd 10 1
198.2.f.e 4 3.b odd 2 1
198.2.f.e 4 33.h odd 10 1
242.2.a.d 2 11.d odd 10 1
242.2.a.f 2 11.c even 5 1
242.2.c.a 4 11.c even 5 2
242.2.c.c 4 11.b odd 2 1
242.2.c.c 4 11.d odd 10 1
242.2.c.d 4 11.d odd 10 2
550.2.h.h 4 5.b even 2 1
550.2.h.h 4 55.j even 10 1
550.2.ba.c 8 5.c odd 4 2
550.2.ba.c 8 55.k odd 20 2
704.2.m.a 4 8.d odd 2 1
704.2.m.a 4 88.l odd 10 1
704.2.m.h 4 8.b even 2 1
704.2.m.h 4 88.o even 10 1
1936.2.a.n 2 44.g even 10 1
1936.2.a.o 2 44.h odd 10 1
2178.2.a.p 2 33.h odd 10 1
2178.2.a.x 2 33.f even 10 1
6050.2.a.bs 2 55.j even 10 1
6050.2.a.ci 2 55.h odd 10 1
7744.2.a.bm 2 88.o even 10 1
7744.2.a.bn 2 88.p odd 10 1
7744.2.a.cy 2 88.k even 10 1
7744.2.a.cz 2 88.l odd 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(22, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} + T^{2} + \cdots + 1$$
$3$ $$T^{4} + 4 T^{3} + \cdots + 1$$
$5$ $$T^{4} + 6 T^{3} + \cdots + 16$$
$7$ $$T^{4} - 2 T^{3} + \cdots + 16$$
$11$ $$T^{4} + T^{3} + \cdots + 121$$
$13$ $$T^{4} + 4 T^{3} + \cdots + 16$$
$17$ $$T^{4} - 2 T^{3} + \cdots + 1$$
$19$ $$T^{4} + 5 T^{3} + \cdots + 25$$
$23$ $$(T^{2} + 2 T - 4)^{2}$$
$29$ $$T^{4} - 10 T^{3} + \cdots + 400$$
$31$ $$T^{4} + 2 T^{3} + \cdots + 16$$
$37$ $$T^{4} + 18 T^{3} + \cdots + 1296$$
$41$ $$T^{4} + 2 T^{3} + \cdots + 361$$
$43$ $$(T^{2} - 3 T - 99)^{2}$$
$47$ $$T^{4} + 8 T^{3} + \cdots + 256$$
$53$ $$T^{4} + 4 T^{3} + \cdots + 256$$
$59$ $$T^{4} - 5 T^{3} + \cdots + 3025$$
$61$ $$T^{4} - 8 T^{3} + \cdots + 256$$
$67$ $$(T^{2} - 11 T - 1)^{2}$$
$71$ $$T^{4} - 8 T^{3} + \cdots + 16$$
$73$ $$T^{4} + 14 T^{3} + \cdots + 17161$$
$79$ $$T^{4} + 30 T^{3} + \cdots + 32400$$
$83$ $$T^{4} + 19 T^{3} + \cdots + 3481$$
$89$ $$(T^{2} + 5 T - 25)^{2}$$
$97$ $$T^{4} + 3 T^{3} + \cdots + 9801$$